When do particles follow field lines?

We examine charged particle transport perpendicular to the large scale magnetic field. We find that the limit of an infinite parallel mean free path of particles diffusing along the large scale magnetic field is a necessary condition for which the diffusive spread of the magnetic field lines leads to a proportional spread of the particles. When it occurs this requires that parallel mean free path is well in excess of the smaller of the system size and the turbulence ultrascale. However, there are alternative situations in which particles may diffuse, but field lines do not. In the latter cases the asymptotic behavior is that which persists after the parallel mean free path exceeds some multiple of the correlation scales. This phenomenon of diffusing particles/non-diffusing field lines is typically determined by the 2D turbulence spectrum, where the diffusion coefficient of the magnetic field due to 2D turbulence can diverge if the spectrum of the 2D fluctuations is not well behaved at small wave numbers. We also show that the classical relation between parallel and perpendicular diffusion for high energy particles is consistent with the field line random walk description of particle diffusion.